The main conjecture of Iwasawa theory relates the algebraic invariants attached to the dual Selmer group of an elliptic curve with the \(p\) -adic L-function, which interpolates values of the complex L-functions. Vatsal [1]} and Greenberg-Vatsal [2]} study congruence properties for complex \(L\) -values as well as for the \(p\) -adic L-functions attached to \(E_1\)  and \(E_2\) . On the algebraic side, the main conjecture in conjunction with work of Schneider [3]} and Perrin-Riou would then predict congruences for the Euler characteristic of the Selmer group.
